Deriving the Magnetic Field from a Magnetic Dipole

[kq6up]

Homework Statement

Show that: ##B=\frac { 1 }{ 4\pi \epsilon r^{ 3 } } \left[ 3\hat { r } \left( \hat { r } \cdot \vec { m } \right) -\vec { m } \right] ##

Homework Equations

B=delxA, m=a*I

The Attempt at a Solution

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I follow my professors derivation. However, she expands the term: ##\vec { \nabla } \times \left( \vec { m } \times \vec { r } \right) =2 \vec m## by just using the determinate. I figured I would go with the less messy "BAC CAB" like she uses on another term. However, I can't get ##2 \vec m## by using BAC CAB. I get ##3 \vec m## instead. I have tried both ways using spherical and cartesian. Does not ##\nabla \cdot \vec m=0##? If I can show that ##\vec { r } \left(\vec \nabla \cdot \vec m \right)=-\vec m##. I would be home free, but I think it should equal zero.

Any E&M or vector calc. gurus, your help would be appreciated.

Thanks,
Chris Maness

 

[TSny]

Due to the product rule for derivatives, BAC CAB doesn't hold here. See #13 at http://www.math.ubc.ca/~feldman/m317/vectorId.pdf

 

[kq6up]

Dang, ok. That was a pretty basic oversight. Thanks

Chris